Kinematics

Speed describes how quickly distance changes

Speed is the distance travelled per unit time. It is a scalar: it has a magnitude but no direction.

Use the totals for the whole journey. Do not take an arithmetic mean of listed speeds unless the time spent at each speed is equal.

The bus spends three times as long on the slow part, so the slow speed accounts for most of the journey time and pulls the average towards 20 km/h.

Velocity adds direction to speed

Velocity is speed in a stated direction. A car travelling at 12 m/s east has a velocity of 12 m/s east; “12 m/s” alone states only its speed. An object can keep the same speed while its velocity changes if its direction changes.

Acceleration is a change per unit time

Acceleration is the change in velocity per unit time. In this course every motion is along one straight line in one direction, so the change in velocity is the final speed minus the initial speed.

1. Write the initial and final speeds in the same units.
2. Calculate final speed minus initial speed. A negative answer means the object is slowing down.
3. Divide by the time interval and state the unit m/s².

Acceleration is not the same as high speed. An aircraft cruising at a steady 250 m/s has zero acceleration because its speed is not changing. A cyclist pulling away from traffic lights at only 3 m/s is accelerating. Ask how much the speed changes each second, not how fast the object is already moving.

Uniform acceleration gives equal speed changes

Uniform acceleration means the velocity changes by the same amount in each equal time interval. For motion in one direction, the speed rises (or falls) by the same amount every second. A uniform acceleration of 2 m/s² means the speed changes by 2 m/s every second.

The speed rises by 2 m/s in every second, so the acceleration is uniform: (11 − 3) ÷ 4 = 2.0 m/s².

Non-uniform acceleration

Acceleration is non-uniform when the speed changes by different amounts in equal time intervals. A cyclist who gains 2 m/s in one second but only 1 m/s in the next is still speeding up, but the acceleration is decreasing.

Everyday motion is often like this. A car pulling away in traffic speeds up quickly at first and then more gradually. A falling skydiver gains speed more and more slowly as air resistance grows, until the speed stops rising.

On a speed–time graph, a straight sloping line has constant gradient and therefore uniform acceleration. A curved line has a changing gradient and therefore non-uniform acceleration.

Plot a motion graph from measured data

1. Put time on the horizontal axis and the measured motion quantity on the vertical axis.
2. Label both axes with quantity and unit, then choose a simple linear scale using most of the grid.
3. Plot every point accurately. Use a line through exact motion records or an appropriate best-fit trend for experimental data.
4. Read coordinates before calculating a gradient; use a large triangle on a straight section.

Distance–time graphs

These Science-course graphs describe motion in one direction. Distance travelled can increase or stay the same; it cannot decrease. The vertical axis records distance, while the horizontal axis records time.

Gradient = speed. The gradient compares the change in distance with the time taken.

• horizontal: at rest
• straight rising line: uniform speed
• changing gradient: non-uniform speed

Reading a distance–time graph

1. 0–4 s: a straight rising line, so the speed is uniform. Gradient = 12 ÷ 4 = 3.0 m/s.
2. 4–7 s: the line is horizontal. The distance stays at 12 m, so the walker is at rest.
3. 7–10 s: a steeper straight line. Gradient = (30 − 12) ÷ (10 − 7) = 6.0 m/s.

Read speed from the gradient, not from the height. At 6 s the walker is 12 m from the start but not moving at all. The graph also never slopes downwards: when motion is in one direction the distance travelled can only stay the same or increase.

Speed–time graphs and distance

Here the vertical axis gives speed directly. Its height tells you how fast the object is moving; its gradient tells you how quickly that speed changes. Gradient = acceleration.

• on the time axis: at rest
• horizontal above zero: uniform speed
• straight slope: uniform acceleration
• curve: non-uniform acceleration

Area under the graph gives distance

Split the region between the graph and time axis into rectangles and triangles. For a speed rising uniformly from 4 m/s to 10 m/s over 3 s, the area is a trapezium:

This method is required here only for uniform speed or uniform acceleration. Because speed is non-negative, every included area contributes positively to distance travelled. Check the units: speed in m/s multiplied by time in s gives distance in m.

Worked example: describe and calculate from a speed–time graph

A trolley accelerates uniformly from rest to 8 m/s in 4 s, travels at 8 m/s for 3 s, then slows uniformly to rest in 2 s.

1. First acceleration = (8 − 0) ÷ 4 = 2.0 m/s².
2. First triangle area = ½ × 4 × 8 = 16 m.
3. Rectangle area = 3 × 8 = 24 m.
4. Final triangle area = ½ × 2 × 8 = 8 m.
5. Total distance = 16 + 24 + 8 = 48 m.

Free fall near Earth

A body is in free fall when gravity is the only significant force on it. Near the Earth's surface, its acceleration is then constant and approximately 10 m/s² downward. The measured value is about 9.8 m/s²; this course uses 10 m/s². “Free fall” does not mean the body must be moving downward.

The speed rises by 10 m/s every second, whatever the mass. A 5 kg steel ball and a 0.1 kg marble dropped together gain speed at the same rate and land together. A heavier body is pulled harder by gravity, but it is also harder to speed up, so its acceleration is the same.

This model assumes air resistance is negligible, which suits a small, dense object falling a short distance. It does not suit a feather, a sheet of paper or a skydiver. For them, air resistance grows as they speed up, so their acceleration falls below 10 m/s² and becomes non-uniform.

Common graph mistakes

“A steeper graph always means greater acceleration.”

Only the gradient of a speed–time graph is acceleration. A distance–time gradient is speed.

“A horizontal speed–time line means the object is stationary.”

It means constant speed. The object is stationary only when the line is on the time axis.

“A higher point on a distance–time graph means a greater speed.”

Height shows how far the object has travelled. Speed is the gradient, so a high horizontal section means the object is at rest.

“A distance–time graph slopes down when the object slows.”

For motion in one direction the distance never decreases. Slowing down makes the graph less steep; stopping makes it horizontal.

“Zero speed means zero acceleration.”

Speed describes the current motion; acceleration describes how velocity is changing.

Worked example: a journey with two stages

A delivery robot travels at a constant 2.5 m/s for 12 s. It then accelerates uniformly from 2.5 m/s to 7.5 m/s for 8 s.

1. First-stage distance = 2.5 × 12 = 30 m.
2. Second-stage distance = average speed × time = ½(2.5 + 7.5) × 8 = 40 m.
3. Total distance = 30 + 40 = 70 m.
4. Total time = 12 + 8 = 20 s.
5. Average speed = 70 ÷ 20 = 3.5 m/s.
6. Acceleration during the second stage = (7.5 − 2.5) ÷ 8 = 0.625 m/s².

Check: the average speed lies between the stage speeds, and m/s divided by s gives m/s².

Connect the complete Kinematics method

A cyclist moves at 4.0 m/s for 3.0 s, then accelerates uniformly to 10.0 m/s over 2.0 s. The acceleration on the sloping speed–time section is (10.0 − 4.0) ÷ 2.0 = 3.0 m/s².

The distance travelled is the area under the speed–time graph: (4.0 × 3.0) + ½(4.0 + 10.0)(2.0) = 26 m. This calculation is separate from free fall: for any object moving under gravity alone near Earth, the acceleration is approximately 10 m/s² downward, even if the object is momentarily moving upward.

Check your understanding

Practise this topic

The topic check is written for this course. Use the feedback to revisit the right explanation, then try a fresh check later.